Method and apparatus for the digitization of and for the data compression of analog signals

ABSTRACT

The invention is based on the idea of providing a method for high-resolution, waveform-preserving digitization of analog signals, wherein conventional scalar logarithmic quantization is transferred to multi-dimensional spherical coordinates, and the advantages resulting from this, e.g., a constant signal/noise ratio over an extremely high dynamic range with very low loss with respect to the rate-distortion theory. In order to make use of the statistical dependencies present in the source signal for an additional gain in the signal/noise ratio, the differential pulse code modulation (DPCM) is combined with spherical logarithmic quantization. The resulting method achieves an effective data reduction with a high long-term and short-term signal/noise ratio with an extremely small signal delay.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority of International Application No.PCT/EP2004/008217, filed Jul. 23, 2004 and German Application No. 103 34357.1, filed Jul. 25, 2003, the complete disclosures of which are herebyincorporated by reference.

BACKGROUND OF THE INVENTION

a) Field of the Invention

The present invention is directed to a method and apparatus for thedigitization of analog signals and a method and apparatus for datacompression of analog signals.

b) Description of the Related Art

In many fields of digital information transmission there exists thecommon goal of transporting an initially analog source signal(measuring, studio-quality audio signal processing, and so on) to areceiver while making use of the advantages of digital transmission inorder to reproduce this analog source signal in the receiver in the formof an analog output signal. For this purpose, the analog signal must bedigitized and quantized on the one hand and a corresponding datacompression must be carried out on the other hand in order to transmitthe signal with the highest possible quality via a digital transmissionsystem with a limited rate. After a PCM encoding, methods for datacompression are usually applied which make use of the redundancy withinthe source signal and irrelevance with respect to specificcharacteristics of the consumer of the signal. The numerous existingmethods for digitizing analog source signals use approaches thatsometimes differ fundamentally, but all of them can be divided into twomain categories:

i) Methods in which the reconstructed waveform very closely approximatesthe original waveform, i.e., there is no use of irrelevance. The term“waveform-preserving” is used here in place of the often applied term“lossless” with respect to the waveform coding because the digitizationof an analog signal at a limited data rate is, on principle, not“lossless” (i.e., infinite entropy of a continuous random variable).

ii) Non-waveform-preserving methods. These methods are of crucialimportance, for example, in the audio field, where often only thesubjective aural impression at the receiver output is decisive (e.g.,use of psychoacoustic masking effects). In this case, waveform changes(amplitude distortion and phase distortion) undergone by the signalthrough quantization and compression generally only play a subordinaterole. Normally, a signal processing using irrelevance leads to areconstructed waveform that differs greatly from the original waveformand, moreover, is often afflicted by extensive signal delay (e.g.,because of spectral transformations or equivalent block-based methods).The transmission quality in a method of the type mentioned above cannotbe measured by a 10 log₁₀ signal/noise ratio (SNR) in the conventionalway; rather, it must be determined by time-consuming listening tests bytrained personnel. However, signal coding methods of this type aretotally unusable for many areas of application (e.g., metrology,recording of waveforms for further signal processing at a later time,real-time signal transmission using digital modulation methods that donot permit significant signal delay, e.g., for wireless digital stagemicrophones).

OBJECT AND SUMMARY OF THE INVENTION

Therefore, it is the primary object of the present invention to providea method and apparatus for the digitization and compression of analogsignals with improved quality.

This object is met by a method for the digitization of analog signalscomprising the steps of digitizing analog source signals, transformingthe digitized source signals from the time domain to the sphericaldomain, wherein the transformation is a D-dimensional transformationwith D>2 and logarithmic quantizing of the radius in the sphericaldomain. It is also met by an apparatus for the digitization of analogsignals comprising means for the digitization of analog source signals,means for the transformation of the digitized source signals from thetime domain to the spherical domain, wherein the transformation is aD-dimensional transformation with D>2, and means for the logarithmicquantization of the radius in the spherical domain. It is further met bya method for the compression of analog signals comprising the steps ofdigitizing analog source signals by the method described above andcarrying out a differential pulse code modulation. It is also met by anapparatus for the compression of analog signals comprising the apparatusdescribed above including an encoder for differential pulse codemodulation and a forward prediction device for determining a startingvalue for the samples of the quantization based on the current state ofa prediction filter, a reconstruction device for reconstructing thesubsequent D samples, a D-dimensional logarithmic spherical quantizationdevice for quantizing the values obtained by the forward prediction inorder to determine a starting cell, wherein the prediction of thedifferential pulse code modulation is run through iteratively in orderto determine a quantization cell with the smallest quantization error.

Accordingly, a method is provided for the digitization of analog sourcesignals in which a D-dimensional spherical logarithmic quantization ofthe analog source signals is carried out.

According to one aspect of the present invention, a method and anapparatus are likewise provided for the compression of analog sourcesignals in which a digitization of analog source signals is carried outin accordance with the method described above, and wherein adifferential pulse code modulation is carried out.

Accordingly, a waveform-preserving method is provided which meets thefollowing requirements: a) low data rate through the use of favorablepacking characteristics of multi-dimensional lattices (vectorquantization) and through the use of dependencies in the sequence ofsamples from the source signal; b) extremely high dynamic range, i.e.,the SNR is constant over a very large modulation range of, e.g., 60 dBor more with respect to the short-term variance of an analog sourcesignal; c) high objectively measurable signal/noise ratio by segmentsfor short segments of samples, preferably within the meaning of theratio of variance of the useful signal to the mean square error; d)insensitivity to specific signal parameters such as probability densityfunction by segment, etc.; and, most importantly, e) introduction of anextremely small signal delay on the order of a few (up to ten) samplingperiods.

The invention is based on the idea of providing a method forhigh-resolution, waveform-preserving digitization of analog signals,wherein conventional scalar logarithmic quantization is transferred tomulti-dimensional spherical coordinates, and the advantages resultingfrom this, e.g., a constant signal/noise ratio over an extremely highdynamic range with very low loss with respect to the rate-distortiontheory. In order to make use of the statistical dependencies present inthe source signal for an additional gain in the signal/noise ratio, thedifferential pulse code modulation (DPCM) is combined with sphericallogarithmic quantization. The resulting method achieves an effectivedata reduction with a high long-term and short-term signal/noise ratiowith an extremely small signal delay.

Further embodiments of the invention are indicated in the subclaims.

In the following, the invention will be described more fully withreference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings:

FIG. 1 shows the proportionality of an arc segment to the radius;

FIG. 2 is a graph illustrating the quantity of quantization cellsassociated with a radius, normalized to the cell quantity per absolutevalue of a sample in scalar quantization;

FIG. 3 is a graph illustrating a signal/noise ratio loss at differentvalues of A;

FIG. 4 is a graph illustrating the signal/noise ratio according to thepresent invention;

FIG. 5 shows a block diagram of a DPCM encoder with backward predictionaccording to the present invention;

FIG. 6 is a graph illustrating the signal/noise ratio according toanother embodiment example of the invention; and

FIG. 7 shows graphs of the signal level and of the signal/noise ratioaccording to another embodiment example of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The aim of the logarithmic quantization is a high dynamic of thequantizer, that is, a wide range of the average signal level in whichthe SNR and, therefore, the maximum relative quantization error

$\frac{\Delta_{q_{i}}/2}{r_{i}}$is constant, wherein Δq_(i) is the width of the ith quantizationinterval and r_(i) is the respective reconstruction value. This aimleads to logarithmic quantization, e.g., within the meaning of the A-lawaccording to K. Tröndle, R. Weiβ, Introduction to Pulse Code Modulation,Oldenbourg Verlag, Munich, 1974. For the range of average modulation,the following signal/noise ratio is given for R>>1 according to N. S.Jayant, P. Noll, Digital Coding of Waveforms, Prentice-Hall, EnglewoodCliffs, N.J., 1984:

$\begin{matrix}{{{10\;{\log_{10}\left( {S\; N\; R} \right)}} = {{{R \cdot 6.02}\mspace{14mu}{dB}} + {10\;\log_{10}\frac{3}{\left( {1 + {\ln(A)}} \right)^{2}}}}},} & (1)\end{matrix}$where R is the average rate of quantization (bits/sample) and Arepresents the usual parameters of logarithmic quantization; that is,the maximum relative error is constant for samples with an amountgreater than 1/A (with reference to a signal range of the quantizationof −1 to +1). Therefore, a signal/noise ratio according to (1) isachieved after an average signal level of the source signal of−B ₁:=20 log₁₀(1/A) dB.   (2)Accordingly, the value B₁ characterizes the dynamic range of thelogarithmic quantization. Also, the signal/noise ratio is entirelyindependent from the probability density function (PDF) of the sourcesignal within this dynamic range so that the method can be useduniversally. However, in comparison with a uniform quantization at fulldrive by an equidistributed signal, the term 10 log₁₀(3/(1+ln(A))²)to acertain extent represents a signal/noise ratio loss by companding, whichis the price of a large dynamic range.

A quantization in spherical coordinates in D dimensions will bedescribed in the following.

Spherical logarithmic quantization belongs to the family of vectorquantization methods (e.g., R. M. Gray, David L. Neuhoff, Quantization,IEEE Transactions on Information Theory, pp. 2325-2383, October 1998,Manfred Herbert, Lattice Quantization of Speech Signals and Speech ModelSignals. Selected Works on Information Systems, No. 79, editor: H.-W.Schüβler, Erlangen 1991). By means of dense spherical packing in manydimensions, considerable gains in signal/noise ratios can be achieved inthe quantization even when no statistical dependencies within thesignals can be made use of. In spherical quantization, a vector x:=(x₁,. . . , x_(D)) with D samples in Cartesian coordinates is expressed inpolar coordinates u:=(φ₁, . . . , φ_(D−1),r). The D−1 angle φ₁ and theradius r are given by the following equations (where j is an imaginaryunit and arg (·) is an argument function which indicates the angle of acomplex number in radian measure):

$\begin{matrix}{\varphi_{1} = {{\arg\left( {x_{1} + {j\; x_{2}}} \right)} \in \left\lbrack {{- \pi},{+ \pi}} \right)}} & (3) \\{{\varphi_{1} = {{\arg\left( \sqrt{{\sum\limits_{l = 1}^{i}x_{l}^{2}} + {j\; x_{i + 1}}} \right)} \in \left\lbrack {{- \frac{\pi}{2}},\frac{\pi}{2}} \right\rbrack}},{i \in {\left\{ {2,\ldots\mspace{11mu},{D - 1}} \right\}.}}} & (4) \\{r^{2} = {\sum\limits_{l = 1}^{D}x_{l}^{2}}} & (5)\end{matrix}$

The reconstruction of the Cartesian components from vector u is carriedout from this according tox _(i) =r·b _(i−1)·sin(φ_(i−1)), i ε {D, . . . , 2}  (6)x ₁ =r·b ₁·cos(φ₁)=r·b ₀,   (7)with the radii b_(i) of the “parallel of latitude” of the unit sphere(radius 1):

$\begin{matrix}{{b_{D - 1} = 1},} & (8) \\{{b_{i} = {1 \cdot {\prod\limits_{l = {i + 1}}^{D - 1}{\cos\left( \varphi_{l} \right)}}}},{i \in \left\{ {{D - 2},\ldots\mspace{11mu},0} \right\}}} & (9)\end{matrix}$

Accordingly, an economical recursive representation is found proceedingfrom φ_(D−1). An efficient implementation of forward transformation andbackward transformation is made possible with little complexity by meansof the so-called CORDIC algorithm (described in Volder, Jack, The CORDICTrigonometric Technique, IRE Trans. Electronic Computing, Vol. EC-8, pp.330-334, September 1959).

Spherical logarithmic quantization is described in the following.

In order to continue to make use of the advantages of logarithmicquantization, i.e., the independence of the SNR from the PDF of thesource signal, with quantization in polar coordinates, the amount(radius) is quantized logarithmically according to the rules of theA-law. The angles φ_(i) are uniformly quantized individually, in eachinstance with the quantization intervals being selected in each instancedepending on the higher-order angles {circumflex over (φ)}_(l),lε{i+1,l+2, . . . , D−1} that have already been quantized and which arealso used for the reconstruction. In this way, a very simpleimplementation of the quantization and the signal reconstruction isachieved and a stepwise processing of the individual coordinates, as inscalar quantization, is preserved by the iterative procedure accordingto (3) to (9).

Due to the proportionality of the length of an arc segment to theradius, the requirements for the logarithmic quantization, i.e., theproportionality of the size of a quantization interval to the signalvalue, are already met for uniform quantization of the angles. FIG. 1shows a simple example for two dimensions D=2. Therefore, the secondterm in (1) must be counted for only one of the total of D dimensions;herein lies the essential reason for the gains achieved by means ofspherical logarithmic quantization.

In order to facilitate the implementation of quantization in sphericalcoordinates, we start from approximately cubic quantization cells, i.e.,the surface of a D-dimensional sphere with radius 1 is quantized by anetwork of D−1 dimensional (hyper-)cubes. As will be shown in thefollowing, however, this suboptimal quantization of the sphericalsurface compared to an optimally densely packed lattice results in aloss of only

${{10\;{\log_{10}\left( \frac{\pi\;{\mathbb{e}}}{6} \right)}} = {1.53\mspace{14mu}{dB}}},$in the limit D→∞, that is, a rate loss of about 1/4 bit/sample, which isaccepted for the sake of facilitated implementation.

The following description explains how the M^(D) (i.e., M:=2^(R))quantization stages available per quantization step are optimallydivided into the individual quantization intervals for the radius andthe surface of the D-dimensional unit (hyper-)sphere.

As is conventional, the applicable compressor characteristic for thelogarithmic range

$\frac{r_{0}}{A} \leq r \leq r_{0}$of the A-law rule is

$\begin{matrix}{{{k(r)} = {r_{0} \cdot \left( {{{c \cdot \ln}\;\frac{r}{r_{0}}} + 1} \right)}},{{{where}\mspace{14mu} c}:=\frac{1}{1 + {\ln\; A}}}} & (10)\end{matrix}$This is derived by

$\begin{matrix}{{k^{\prime}(r)}:={\frac{\mathbb{d}{k(r)}}{\mathbb{d}r} = {\frac{c \cdot r_{0}}{r}.}}} & (11)\end{matrix}$

A refers to the free parameter of the A-law and r₀ is introduced as anormalization factor for the radius which will be described more fullyin the following. With M_(D) quantization intervals for the radius (Dthcomponent of vector u), the width of the quantization cell in directionof the radius is given by:

$\begin{matrix}{{{\Delta\;{r(r)}} \approx \frac{r_{0}}{M_{D} \cdot {k^{\prime}(r)}}} = {\frac{1}{M_{D} \cdot c} \cdot {r.}}} & (12)\end{matrix}$It will be noted that Δr(r) is not dependent on r₀ in the domain inquestion.

In order that a saturation behavior similar to that in Cartesiancoordinates can be obtained with quantization in polar coordinates, themaximum value r₀ for the radius is normalized in such a way that theD-dimensional sphere has the same volume as a D-dimensional cube withedge length 2 (corresponding to a quantization range x_(i)ε[−1;1] inevery dimension), i.e.,

${V_{sphere}\overset{!}{=}{{V_{cube}\text{:}\mspace{14mu}{\alpha_{D} \cdot r_{0}^{D}}}\overset{!}{=}2^{D}}},$with the volume α_(D) of the D-dimensional unit sphere according to J.H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups,Springer-Verlag, 3rd edition.

$\begin{matrix}{{\alpha_{D} = \frac{\pi^{D/2}}{\left( {D/2} \right)!}}{{Therefore}\text{:}}} & (13) \\{{r_{0} = \frac{2}{\alpha_{D}^{1/D}}},{{{where}\mspace{14mu} r_{0}} > {1\mspace{14mu}{\forall{D \in {I\;{N.}}}}}}} & (14)\end{matrix}$

To facilitate the determination of the resulting signal/noise ratio, r=1is used in the following, i.e., those quantization cells lying on thesurface of a sphere with unit radius are considered. This is possiblewithout limiting generality because the SNR is not dependent upon radiusdue to the logarithmic quantization in the domain:

$\frac{r_{0}}{A} \leq r \leq {r_{0}.}$The width of the quantization cells is now given by:

$\begin{matrix}{\Delta:={\frac{1}{M_{D} \cdot c}.}} & (15)\end{matrix}$

By means of the uniform angular quantization mentioned above, thesurface of the D-dimensional unit sphere is divided into M_(φ) cellshaving in each instance the form of D−1-dimensional cubes. For thesurface of this D−1-dimensional sphere, according to J. H. Conway and N.J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag,3rd edition:S=β _(D) ·r ^(D−1), where β_(D) =D·α _(D).   (16)

The division of the quantizing stages into radius and angle is describedthe following.

The following considerations must be taken into account for a fairdivision of the M_(D) available quantization cells into radius andsurface of the unit sphere: with respect to the approximation by cubicquantization cells, i.e., approximately equal dimensioning of thequantization cells in all dimensions (radius and arc segment), thesurface of the unit sphere (r=1) must be divided into M_(φ) equal(D−1)-dimensional cubes with identical edge length Δ.

$\begin{matrix}{S = {\beta_{D}\overset{!}{=}{M_{\varphi} \cdot {\Delta^{D - 1}.}}}} & (17)\end{matrix}$Since there are M=2^(R) available intervals per sample,

$\begin{matrix}{{M_{\varphi} \cdot M_{D}}\overset{!}{=}M^{D}} & (18)\end{matrix}$must be demanded in addition.

With (15), (17) and (18), the quantity of the quantization intervalsavailable for the quantization of the radius is given by:

$\begin{matrix}{M_{D}:={{M \cdot \frac{1}{\beta_{D}^{1/D}}}{\frac{1}{c^{{({D - 1})}/D}}.}}} & (19)\end{matrix}$

The quotients M_(D)/(M/2) of the quantity of quantization cellsdedicated to the radius are plotted in FIG. 2 normalized to the cellquantity per absolute value of a sample in scalar quantization (D=1)compared to the dimension number for a series of examples of parameter Aof A-law companding. The values of A are selected in such a way that theuse of A-law companding leads to very small quantization intervals whichare smaller by a factor of 2^(Δn), ΔnεIN than they would be with uniformquantization with an identical total interval number:

$\begin{matrix}{{{\Delta\;{r\left( {r \leq {1/A}} \right)}} = {2^{{- \Delta}\; n} \cdot \frac{1}{M_{D}}}},{i.e.},} & (20) \\{2^{\Delta\; n} = {\frac{A}{1 + {1\; n\; A}}.}} & (21)\end{matrix}$Consequently, the resolution is increased by Δn bits for small values ofr.

FIG. 2 shows that spherical logarithmic quantization beyond a factor of8 allocates more intervals to the radius than a scalar quantization withan equivalent resolution in the range of very small signal values.Therefore, a gain of 3 bits/sample or 18 dB for spherical logarithmicquantization is achieved in this range.

The edge length of the quantization cells on the surface of the unitsphere is obtained using (15):

$\begin{matrix}{\Delta = {\frac{1}{M} \cdot {\left( \frac{\beta_{D}}{c} \right)^{\frac{1}{D}}.}}} & (22)\end{matrix}$The constant c is determined by the selected dynamic range, see (2) and(10).

The actual uniform quantization of the angle variables for identical arcsegments on the unit sphere is obtained by means of the recursiveequations (8) and (9). M_(i) intervals are available for the coordinateφ_(i):

$\begin{matrix}{M_{D - 1} = \left\lfloor \frac{\pi}{\Delta} \right\rfloor} & (23) \\{{M_{i}\left( {{\hat{\varphi}}_{l + 1},\ldots\mspace{11mu},{\hat{\varphi}}_{D - 1}} \right)} = {{\left\lfloor \frac{\pi \cdot {\hat{b}}_{i}}{\Delta} \right\rfloor\mspace{14mu}{for}} \in \left\{ {{D - 2},{D - 3},\ldots\mspace{11mu},2} \right\}}} & (24) \\{{{M_{1}\left( {{\hat{\varphi}}_{2},\ldots\mspace{11mu},{\hat{\varphi}}_{D - 1}} \right)} = \left\lfloor \frac{2\;{\pi \cdot {\hat{b}}_{1}}}{\Delta} \right\rfloor},} & (25)\end{matrix}$where └x┘εIN is the greatest whole number≦x with xεIR⁺ and {circumflexover (b)}_(i) corresponding to (8) and (9) for quantized angles{circumflex over (φ)}_(i).

It should be noted that the quantity M_(i) of quantization intervals indimension i is a function of the quantization cell selected indimensions i+1, . . . ,D−1 and, therefore, can be calculated iterativelystarting from M_(D−1). It is not possible to calculate in advance.

The assignment of an index Nε{0,1, . . . ,M^(D)−1} to the currentquantization cell and the reconstruction can be carried out using linkedlookup tables as is known from shell mapping, e.g., according to RobertF. H. Fischer, Precoding and Signal Shaping for Digital Transmission,pp. 258-281, John Wiley & Sons, Inc., New York, 2002, ISBN 0471 22410 3:

The assignment of an index to the radius is not dependent upon the cellindex on the unit sphere so that the problem of index assignment for thesurface of the unit sphere can now be considered.

The starting number of the index is 0, the maximum ending number of theindex is M_(φ)−1 corresponding to the quantity of quantization cells onthe surface of the unit sphere. M_(D−1) is the quantity of subspheres ofdimensionality D−1 and can be calculated from (23). A first-order lookuptable with M_(D−1) entries N₀, . . . , N_(M) _(D−1) ⁻¹ is used for thesake of fast implementation. In this connection, N_(i) designates thesmallest index of all cells belonging to the ith quantization intervalfor the angle {circumflex over (φ)}_(D−1):

$\begin{matrix}{{N_{0} = 0}{{N_{i + 1} = {\sum\limits_{v = 0}^{i}{M_{D - 2}\left( {{\hat{\varphi}}_{D - 1} = {v \cdot \frac{\pi}{\Delta}}} \right)}}},{i \in \left\{ {0,\ldots\mspace{11mu},\left( {M_{D - 1} - 2} \right)} \right\}}}} & (26)\end{matrix}$

Accordingly, there are N_(i+1)−N_(i) cells for a fixed ith value of{circumflex over (φ)}_(D−1). For every N_(i), a second-order lookuptable is applied which contains the indices O_(ij) which again designatethe smallest index of all of the cells of the jth quantization intervalfor the angle {circumflex over (φ)}_(D−2) within the D−1-dimensionalsubsphere to which N_(i) refers.

$\begin{matrix}{{O_{i,0} = 0}{{O_{i,{j + 1}} = {\sum\limits_{v = 0}^{j}{M_{D - 3}\left( {{{\hat{\varphi}}_{D - 2} = {v \cdot \frac{\pi}{\Delta}}},{{\hat{\varphi}}_{D - 1} = {i \cdot \frac{\pi}{\Delta}}}} \right)}}},{i \in \left\{ {0,\ldots\mspace{11mu},\left( {M_{D - 2} - 1} \right)} \right\}},{j \in {\left\{ {0,\ldots\mspace{11mu},\left( {{M_{D - 2}\left( {{\hat{\varphi}}_{D - 1} = {i \cdot \frac{\pi}{\Delta}}} \right)} - 2} \right)} \right\}.}}}} & (27)\end{matrix}$

This procedure leads to D−2 orders of linked lookup tables in order toindicate the D−1 angles (the D−1-order lookup table for angle{circumflex over (φ)}₁ would contain consecutive cell indices and musttherefore not be tabularized).

The quantization noise and the signal/noise ratio are considered atgreater length in the following.

As long as there is a sufficiently large quantity of quantization cellsin D dimensions, the usual approximation of an equidistributedquantization error can be applied within these cubic cells, each cellbeing represented by its center. The output of the quantization noise isgiven approximately by:

$\begin{matrix}{\frac{\Delta^{2}}{12} \cdot {D.}} & (28)\end{matrix}$

The spherical logarithmic quantization forces Δ(r)=Δ·r in all dimensionsfor

$\frac{r_{0}}{A} \leq r \leq {r_{0}.}$Therefore,

$\begin{matrix}{{{S\; N\; R} = {\frac{r^{2}}{\Delta^{2} \cdot r^{2} \cdot \frac{D}{12}} = {{F(D)} \cdot M^{2}}}},} & (29) \\{{{{where}\mspace{14mu}{F(D)}}:={\frac{12}{D} \cdot \left( \frac{c}{\beta_{D}} \right)^{\frac{2}{D}}}},} & (30)\end{matrix}$see also (22). As is wanted, the SNR is independent from the variance ofthe signal in this range.

Inserting (13) and (16) gives:

$\begin{matrix}{{F(D)} = {\frac{12}{\pi} \cdot \frac{1}{D^{\frac{D + 2}{D}}} \cdot \left( {\left( {D/2} \right)!} \right)^{2/D} \cdot {c^{2/D}.}}} & (31)\end{matrix}$

When (29) is considered, F(D) can be interpreted as a loss with respectto the rate-distortion bound for iid Gaussian random variables (6dB-per-bit rule). This relationship is shown in FIG. 3 for differentvalues of A.

Stirling's approximation

${{x!} \approx {\sqrt{2\;\pi\; x}\left( \frac{x}{\mathbb{e}} \right)^{x}}},$is used to calculate the limit

$\begin{matrix}{{\lim\limits_{D\rightarrow\infty}\;{F(D)}} = {\frac{6}{\pi\;{\mathbb{e}}}\hat{=}{{- 1.53}\mspace{14mu}{dB}}}} & (32)\end{matrix}$representing the loss in relation to the rate-distortion bound which iscaused by the suboptimal cubic quantization cells (relative to(D−1)-dimensional hyperspheres for the quantization of the surface of aD-dimensional unit sphere and D→∞). In other words, sphericallogarithmic quantization makes it possible to compensate again for theloss described in (1) resulting from companding to only 1.53 dB.Therefore, through the selection of the parameter A and D, it ispossible in theory to achieve a dynamic range of any size (range ofconstant signal/noise ratio over the average signal level) withouthaving to accept significant losses in the maximum signal/noise ratiothat can be achieved. The asymptotic SNR loss of 1.53 dB corresponds toa rate loss of 1/4 bit/sample which is acceptable for the sake of anextreme reduction in complexity.

In view of the fact that no knowledge of a probability density functionfor iid source signal values can be deliberately assumed for the purposeof logarithmic companding, the rate-distortion function for the Gaussianrandom variables again gives the lower bound for the achievable rate (ordistortion) at a given distortion (or rate) (Berger's upper bound of therate-distortion function according to T. Berger, Lossy Source Coding,IEEE Transactions on Information Theory, pp. 2693-2723, October 1998).Therefore, improvements going beyond 1.53 dB or 1/4 bit/sample are notpossible anyway under the present requirements and restrictions.

FIG. 4 shows, by way of example, the distance of the signal/noise ratiofrom the rate-distortion bound R·6 dB depending in the average signallevel (10 log 10 (variance)) of iid random variables in Gaussiandistribution when A=48270 for different dimension numbers of sphericallogarithmic quantization. These simulation results exactly satisfy thetheoretical analysis according to FIG. 3. At first glance, the verylarge values for A that are used in the examples in FIGS. 3 and 4 and inthe following examples seem unrealistic and impossible to implement.However, it should be taken into account that substantially more than Mintervals are usually used by means of (19) for the quantization of theradius (M_(D)>M/2) (see FIG. 2). Accordingly, even for very low rates,e.g., R=4 bits/sample, there are very fine quantization intervals forthe radius. Further, (10) is an invertible function for every value ofA>1 and is therefore very well suited in every case for specifying adetermined non-uniform quantizer. This approach for proper waveformcoding with extremely low signal delay should preferably not be used forrates below 3 bits/sample.

Further, it is clear from FIG. 4 that the dynamic range rises sharplythrough two effects as the dimension number increases with a constantparameter A. For one, the normalization radius r₀ increases; foranother, the limiting of the logarithmic compression acts on only onedimension, the radius, whereas for D−1 dimensions the proportionality ofthe expansion of the quantization cell to the signal value (=arc lengthto radius with fixed angle difference) is also retained for very smallsignal values. Therefore, the dynamic range in D dimensions expandsapproximately toB _(D) ≈B ₁+20 log₁₀(r ₀)+10 log₁₀(D).   (33)

In addition, because of the averaging effect within D values, thesaturation strength increases so that a further increase in the dynamicrange is achieved. There is an infinitely large dynamic range for eachdesired value of A in limit (D→∞).

As in every vector quantization method, the delay of sphericallogarithmic quantization by nature of its structure is equal to exactlyD samples. As can be seen from FIGS. 3 and 4, the majority of thepossible gains is already achieved at very small values of D (up to 5).

The combination of a spherical quantization and DPCM is described in thefollowing.

Correlations between the samples q[k] are made use of efficiently bymeans of differential PCM (DPCM) (see FIG. 5). A prediction error signalx[k] is generated by subtracting predicted samples that are obtained bymeans of a linear predictor filter H_(p)(Z)·z⁻¹ from reconstructedsamples {circumflex over (q)}[k]. For an ideal predictor, the predictionerror sequence x[k] shows a white power density spectrum (PSD) andminimal variance. This predictor filter is usually adaptive in order toadapt to a nonstationary source. With the aim of describing theinteraction with spherical logarithmic quantization as simply aspossible, only examples for a fixed predictor filter will be presentedin the present case. In audio signals, for example, even very short,fixed prediction error filters (designed with respect to a compromisecriterion) usually offer gains of more than 18 dB with a samplefrequency of 44.1 kHz. Even segmentwise gains averaged over 6000 samples(0.136 s) of less than 15 dB can be observed only very rarely (see FIG.7 compared to FIG. 3). Further, the gains are limited by adaptiveprediction in case the signal delay is limited to a few samples andnoticeable effects which are produced by updating the coefficients mustbe prevented. Of course, the following material can be generalized so asto apply directly to adaptive prediction.

It should be noted that a gain in SNR expressed by a reduction in themean square error between the original samples and the reconstructedsamples is possible when using the so-called backward prediction shownin FIG. 5.

With logarithmic quantization, the SNR of the prediction error signalx[k] is not dependent on its variance or, to express this differently,the output of the quantization noise is proportional to the signaloutput. Accordingly, the prediction gain, i.e., the quotient of thevariances of q[k] and x[k] can be converted directly to a gain in SNR.Therefore, logarithmic quantization is a good choice for DPCM. Inaddition to this, no further signal delay inherent to the system iscarried out by DPCM compared with PCM because an optimal predictionerror filter for maximum prediction gain is fundamentally causal andstrictly minimum phase (see L. Pakula and S. Kay, Simple Proofs of theMinimum Phase Property of the Prediction Error Filter, IEEE Transactionson ASSP, Vol. 31, 1983) and can therefore be inverted withoutstructure-induced delay.

The gradient descent method will be described more fully in thefollowing.

Application of spherical logarithmic quantization to DPCM with backwardprediction is subject to the same set of problems as any other vectorquantization method: in order to calculate the current prediction errorsignal x[k], all of the preceding reconstructed samples q[k−i], i=1, 2,. . . must be present. For a high prediction gain, the immediatelypreceding values (i=1, i=2) in particular are indispensable.Unfortunately, this requirement cannot be reconciled with thequantization of blocks of length D samples.

Any of the methods mentioned in the literature for combining vectorquantization with DPCM can be used to solve this problem. In the presentcase, a method relying on the principle of analysis by synthesis isused. This method is known, for example from CELP waveform codingmethods according to N. S. Jayant, P. Noll, Digital Coding of Waveforms,Prentice-Hall, Englewood Cliffs, N.J., 1984, and this approach iscombined with a discrete gradient descent method.

In order to resolve the conflict between DPCM and vector quantization,the square Euclidean distance between vectors of samplesq[l]=(q[l·D],q[l·D+1], . . . ,q[l·D+D−1])   (34)and a corresponding reconstruction vector {circumflex over (q)}[l] isminimized l=└k/D┘.

It should be noted at this point that, in addition to sphericallogarithmic quantization, the calculation of the correspondingprediction error signal and the inversion of the prediction error filtermust be included in the calculation of a pair q,{circumflex over (q)},wherein preceding reconstruction vectors {circumflex over(q)}[l−m],m=1,2, . . . are resorted to, but themselves remain unchanged.The aim of the algorithm is to find those quantization cells for x[l]for which the metric

$\begin{matrix}{{d^{2}\left( {q,\hat{q}} \right)} = {\sum\limits_{i = 0}^{D - 1}\left( {{q\left\lbrack {{D \cdot l} + i} \right\rbrack} - {\hat{q}\left\lbrack {{D \cdot l} + i} \right\rbrack}} \right)^{2}}} & (35)\end{matrix}$is minimized.

In order to find a suitable starting value for the algorithm, we suggestbeginning with a forward prediction for the current D samplescorresponding to a deactivation of the chain from ADC and DAC in FIG. 5,or inputting q[k], k=1·D,l·D+1, . . . ,l·D+D−1 directly into thepredictor filter (in FIG. 5) instead of {circumflex over (q)}[k].Spherical logarithmic quantization is now carried out on the resultingvector x and a starting vector y is generated which in turn gives avector {circumflex over (q)} by conventional inversion of the predictionerror filter (DPCM receiver structure). In this way, the metriccalculation for a given quantized vector y_(i) can be carried out.Starting from an actual reconstruction vector y_(i), the reconstructionvectors of all 2D nearest neighbors Y_(j[i]) in D dimensions aredetermined and the associated metrics are calculated with (35) andcompared to one another. The vector with the smallest metric is used forthe next iteration, i.e., arg min_(j)d²(q,{circumflex over (q)}_(j[i]))generates an updating of i, wherein {circumflex over (q)}_(i)corresponds to the quantized vector y_(i) and is calculated by invertingthe prediction error filter. If no such vector exists, the algorithm isterminated and gives the index of the vector y_(i) to be transferred. Itshould be noted here that the neighbor cells usually have very differentindices and it is not a trivial matter to identify these cells. When weconsider D=3, for example, the angles of the azimuth (φ₁) are quantizedmore crudely at large elevations (φ₂ near ±π/2) than at small elevations(φ₂ near 0) because the latter have a greater parallel of latitude onthe sphere surface. For fast implementation, the indices of the neighborcells could be stored in a ROM for all cells provided M_(φ) is not toolarge.

Since linear prediction is included in the optimizing process, theresulting SNR is often greater than can be predicted by adding the gains(in dB) from spherical logarithmic quantization and prediction (DPCM).Noteworthy gains in SNR can be observed particularly in case of smallrates (e.g., R<5 bits/sample), which will be described in the followingand with reference to FIG. 6. The Author is not currently aware of ananalytic result for the SNR that can be achieved through this algorithm.

Simulations show that the required number of iterations is on averageapproximately 0.25 (per D sample) and, when limited to a maximum of 3iterations, no significant losses are observed compared to an unlimitedsearch space, so that the search for the optimal quantization cell forlow dimension numbers can certainly be implemented in real time.Further, small values of D already yield large gains (see FIGS. 3 and6).

On the reception side, there is no change compared with conventionalDPCM methods. It should be noted here that the total delay of thetransmission system is only D samples and it is excellently suited totransmissions which must meet the requirements of an extremely shortdelay.

Variants of the method are described more fully in the following.

To speed up the search for the most favorable quantization leading tominimum distortion, e.g., according to (35), all algorithms of thelattice decoding are applicable in principle or for finding a maximumlikelihood codeword in channel coding (see, for example, Erik Agrell,Thomas Eriksson, Alexander Vardy, Kenneth Zeger, Closest Point Search inLattices, IEEE Transactions on Information Theory, pp. 2201-2214, August2002, and the references cited therein). This method is preferablytransformed into spherical coordinates.

A variant without iterative determination of the quantization cell isprovided in that a non-uniform quantization of the angle coordinates iscarried out instead of taking into account statistical dependencieswithin the D signal values which are actually to be quantized by meansof a linear prediction filter according to FIG. 5 (DPCM). In thisconnection, the linear statistical links of the D signal values whichare actually to be quantized to preceding signal values are made use ofby means of conventional DPCM with backward prediction. That is, thereis a backward prediction in which an updating of the prediction filterby D steps is carried out after D signal values.

It is suggested that the logarithmic quantization of the amount ismaintained so that the signal points can be normalized subsequently insuch a way that they lie on a sphere with radius 1. This allows theanalytic calculation of the probability density function of the signalpoints on the spherical surface, e.g., assuming a Gaussian signalprocess from the autocorrelation function of the source signal or adirect empirical determination of relative occurrences of the signalpoints on the spherical surface.

A non-uniform cubic quantization of the surface of a sphere with radius1 can be determined in this way for M>>1, e.g., by means of theoptimization formula according to (9). This formula states that, on theaverage, the amount of every quantization should be equal to thequantization noise. When Δ_(i) is the edge length of a (D−1)-dimensional cubic quantization cell and z_(i) is the associated centerpoint, that is, the reception-side reconstruction vector with amount 1,and ƒ(v);v:=(φ₁, . . . ,φ_(D−1)) is the probability density function orrelative occurrence of signal values on the spherical surface, thefollowing equation should be approximately satisfied:

$\begin{matrix}{{\left( {D - 1} \right){\frac{\Delta_{i}^{2}}{12} \cdot {f\left( z_{i} \right)} \cdot \Delta_{i}^{D - 1}}} = {{const}.}} & (36)\end{matrix}$with the secondary condition

$\begin{matrix}{{\sum\limits_{i = 1}^{M\;\varphi}\Delta_{i}^{D - 1}} = {\beta_{D}.}} & (37)\end{matrix}$

This leads directly to a (D−1)-dimensional compressor function k(v) bywhich the non-uniform quantization of the sphere surface is uniquelydetermined. The non-uniform quantization can then be completed, forexample, as in the one-dimensional case, by a nonlinear deformation ofvector v to a vector z:=k(v), subsequent uniform quantization to vectorz_(i) according to III-C, and application of the inverse function{circumflex over (v)}:=k⁻¹ (z_(i)) subsequently for obtaining thereconstruction vector.

The resulting (D−1)-dimensional compressor function should preferably beapproximated by an analytically describable function or, analogous tothe usual representation in one-dimensional compression by means ofstraight-line segments (e.g., 13-segment characteristic line, see [5]),by (D−2)-dimensional partial planes (with constant partial derivations).When this approximation is latticed orthogonally over the space

$\left( {{- \frac{\pi}{2}},\frac{\pi}{2}} \right\rbrack^{D - 2} \times \left( {{- \pi},\pi} \right\rbrack$over the D−1 angles, the compressor function can be evaluated in asimple manner.

To simulate spherical logarithmic quantization in combination with DPCM,the quantization of the overture and the aria “Der Vogelfänger bin ichja” from the opera “Zauberflöte” by Wolfgang Amadeus Mozart was examined(Philips Classics Productions 1994 (DDD), Mozart, “Der Vogelfänger binich ja” (Die Zauberflöte), Polygram Records #442569-2, track 3).

FIG. 6 shows simulation results for R=3, R=4 and R=7 bits/sample whenA=102726, A=48270, A=4858 and with different dimension numbers D.

The signal was first coded, then decoded and the SNR was calculated bycomparison with the original CD signal. The signal/noise ratio isaveraged over the entire piece of music. At the low rates, the SNR risesmore steeply with D compared to FIG. 3. The reason for this is animproved averaging. The different values of parameter A must be takeninto consideration when comparing the results. The aria “Vogelfänger” isstored on audio CD with an average signal level of −32.15 dB and offersan excellent example for audio signal coding because of the signaldynamics and tone colors (prelude, singing, panpipe). These simulationresults can be regarded as representative for a large number of audiosignals because a universal predictor filter of low order (P=2) wasused.

FIG. 6 shows the measured signal/noise ratios averaged over the entirepiece of music. It is apparent from a comparison of FIG. 3 (or (1)) andFIG. 6, where D=1, R=7, that the average prediction gain for this simplepredictor is about 20 dB to 23 dB (sample frequency: 44.1 kHz!). WhenR=7, the total gain in all values of D can be favorably approximated bythe sum of the two individual gains from spherical logarithmicquantization and DPCM; when R=4, this applies only for D≧5. (Of course,(1) is not applicable, for example, for R=3 or R=4 and D=1, A=48270.)

The average signal level of the overture is around −27.20 dB; this pieceof music is characterized by a very large dynamic range from below −70dB to −17 dB (see also FIG. 7). The segment signal level and the SNRsfor the overture example are plotted in FIG. 7. Each segment comprises6000 samples and therefore corresponds to a time range of 0.136 s. Inthis case, spherical logarithmic quantization is applied in D=3dimensions with A=1014 and A=4858 and in D=6 dimensions with A=48270 andwith the same compromise predictor of the order of P=2 as in the firstexample. The top curve shows the large dynamics brought about by theknown loud tones divided by pauses over a plurality of bars at the startand in the middle of this popular piece of music. The curves for A=1014and A=4858 disclose the advantage of an increasing A with respect to theSNR by segment, wherein the pauses should be noted in particular. Thebottom curve in FIG. 7 demonstrates this aspect for a further increasein A and the resulting further gain in SNR of 3 dB because of the higherdimension number D=6 (see also FIG. 3). It should be noted that in spiteof the low rate of R=4 bits/sample, a 10 log₁₀(SNR)>35 dB (with respectto the original CD) is maintained even in the pauses.

The value A=48270 corresponds to a compression of 12 bits, e.g., forD=1, R=12 a resolution of 24 bits is achieved for small signal values.Of course, there is never a resolution corresponding to 16 bits for D=1,R=4 because the interval number is too low. However, when sphericallogarithmic quantization is applied in 6 dimensions, the extremely largevalue of A=48270 is actually well chosen for a high minimum value of theSNR. In this case, the resolution is even identical to that of theoriginal CD data for segments with a signal level of −70 dB and lower.To date, no one has been found, even among trained personnel, who wascapable of reliably hearing a difference between the quantized signal atR=4, D=6, A=48270 and the original signal.

The preceding description shows a waveform-preserving digitizationmethod for analog source signals which, on the one hand, combines thegain by multidimensional quantization with the advantages of logarithmicquantization and, on the other hand, is capable of additionallyincreasing the objective signal/noise ratio through the addition ofprediction gains. In conclusion, it is noted that, in addition to afavorable exchange of rate and distortion, this method is distinguishedabove all by an extremely large dynamic range accompanied at the sametime by an extremely low system-induced delay of the signal by only afew sample intervals.

While the foregoing description and drawings represent the presentinvention, it will be obvious to those skilled in the art that variouschanges may be made therein without departing from the true spirit andscope of the present invention.

1. A method for processing digital source signals comprising thefollowing steps: digitizing analog source signals; transforming thedigitized source signals from the time domain to a spherical domain,wherein the transformation is a D-dimensional transformation with D>2;and logarithmic quantizing the radius in the spherical domain.
 2. Themethod according to claim 1, wherein a transformation of the sourcesignals into spherical coordinates is carried out.
 3. The methodaccording to claim 2, wherein a logarithmic quantization of the amountis carried out in spherical coordinates.
 4. The method according toclaim 2, wherein a non-uniform quantization of the angle coordinates ofthe spherical coordinates is carried out.
 5. The method according toclaim 1, comprising the steps of quantizing the analog source signals inpolar coordinates, wherein the amount of the source signals in polarcoordinates is logarithmically quantized, and wherein the quantizationof the angle in polar coordinates of the analog source signals iscarried out depending on the quantized angle of higher order.
 6. Themethod according to claim 5, wherein the quantization of a D-dimensionalsphere with a unit radius is carried out by a network of D−1-dimensionalcubes.
 7. The method according to claim 5, wherein those quantizationcells which lie on the surface of a sphere with unit radius areselected, wherein the quantization cells are D−1-dimensional cubes. 8.The method for the compression of analog signals, comprising the stepsof carrying out a digitization of analog source signals by a methodaccording to claim 1, and carrying out a differential pulse codemodulation.
 9. The method according to claim 8, comprising the steps ofcarrying out a forward prediction for determining a starting value forsamples of the quantization based on the current state of a predictorfilter, reconstructing the subsequent D samples, carrying out aD-dimensional logarithmic spherical quantization of the values obtainedby the forward prediction in order to determine a starting cell,iterative run-through of the prediction of the differential pulse codemodulation in order to determine a quantization cell with the smallestquantization error.
 10. The method according to claim 8, wherein alattice decoding is carried out for determining a favorable quantizationcell.
 11. The method according to claim 8, wherein a backward predictionof a differential pulse code modulation is carried out, wherein anupdating of the prediction filter by D steps is carried out afterprocessing D samples.
 12. An apparatus for processing digital sourcesignals comprising: means for the digitization of analog source signals;means for the transformation of the digitized source signals from thetime domain to a spherical domain, wherein the transformation is aD-dimensional transformation with D>2; and means for the logarithmicquantization of the radius in the spherical domain.
 13. Apparatusaccording to claim 12, with a transformation device for carrying out atransformation of the analog source signals in spherical coordinates.14. Apparatus according to claim 12, wherein a logarithmic quantizationof the amount is carried out.
 15. Apparatus according to claim 13,wherein a non-uniform quantization of the angle coordinates of thespherical coordinates is carried out.
 16. The apparatus of claim 12,wherein the apparatus further comprises: an apparatus for thecompression of analog signals and an encoder for differential pulse codemodulation.
 17. Apparatus according to claim 16, with a forwardprediction device for determining a starting value for the samples ofthe quantization based on the current state of a predictor filter, areconstruction device for reconstructing the subsequent D samples, aD-dimensional logarithmic spherical quantization device for quantizingthe values obtained by the forward prediction in order to determine astarting cell, wherein the prediction of the differential pulse codemodulation is run through iteratively in order to determine aquantization cell with the smallest quantization error.
 18. Apparatusaccording to claim 17, wherein a lattice decoding device is carried outfor determining a favorable quantization cell.
 19. Apparatus accordingto claim 17, wherein a backward prediction of a differential pulse codemodulation is carried out, wherein the prediction filter is updated by Dsteps after processing D samples.